You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. Your work must be expressed in standard mathematical notation rather than calculator syntax.

Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.

x -3 0 3 6
f(x) -5 4 1 7
f'(x) -1 2 -2 4

The table above gives values of a twice-differentiable function f and its first derivative f' for selected values of x. Let g be the function defined by

g(x)= f(x^3-x )

Required:
What is the value of g (3)?

dunnodunnodunnodunnodunnodunnodunnodunnodunnodunnodunnodunnodunnodunno

dunno

Answer:

We can use the formula for continuous compound interest to find the balance in Joseph and Deb's savings account after 1 year:

A = Pe^(rt)

where A is the balance, P is the principal (initial deposit), e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate as a decimal, and t is the time in years.

Substituting the given values, we get:

A = $600.00e^(0.05*1)

Using a calculator, we get:

A ≈ $632.57

Therefore, Joseph and Deb will have approximately $632.57 in their savings account after 1 year. They can spend up to this amount on their trip. Rounded to the nearest cent, the answer is $632.57.

Using the graph we can see that the value is:

f(-3) = 6

(or f(3)= 7, depengin on which you need).

so here we have the graph of f(x), to find the value of f(-3), you first need to find x = -3 on the horizontal axis.

Then move vertically until you meet the graph of the function, once you do that, you can move horizontally until you meet the vertical axis, and at that point you can identifty the value of the function.

We can see that:

f(-3) = 6

(if instead you wanted f(3), it is equal to 7, but it wasn't in the options)

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The prove of the given trigonometric function is given below.

The given trigonometric function is,

(sin⁴(x) - cos⁴(x))/(sin²(x) - cos²(x))

Now proceed left hand side of the given expression:

We can write the expression  as,

⇒[(sin²(x))² - (cos²(x))²]/(sin²(x) - cos²(x))

Since we know that ,

Algebraic identity:

a² - b²  = (a-b)(a+b)

Therefore the above expression be

⇒(sin²(x) - cos²(x))(sin²(x) + cos²(x))/(sin²(x) - cos²(x))

⇒(sin²(x) + cos²(x))

Since we know that,

Trigonometric Identities come in handy when trigonometric functions are used in an expression or equation. Trigonometric identities hold for all values of variables on both sides of an equation. Geometrically, these identities include one or more trigonometric functions (such as sine, cosine, and tangent).

Then,

sin²(x) + cos²(x)² = 1 is an trigonometric identity

Hence,

(sin⁴(x) - cos⁴(x))/(sin²(x) - cos²(x)) = 1

Hence proved.

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The value of slope of the secant line is, 6

We have to given that;

The graph of line are shown in figure.

Now, Take two points on the secant line is,

⇒ (0, 0) and (0.1, 0.6)

Hence, The slope of the secant line is,

m = (0.6 - 0) / (0.1 - 0)

m = 0.6 / 0.1

m = 6/1

m = 6

Thus, The value of slope of the secant line is, 6

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\(f(x)=2(3)^{x+1} \\\\[-0.35em] ~\dotfill\\\\ f(2)=2(3)^{(2)+1}\implies f(2)=2(3)^3\implies f(2)=2(3^3) \\\\\\ f(2)=2(27)\implies f(2)=54\)

Answer:

5⁰ × 7⁰ × 3⁰ = 1

Step-by-step explanation:

5⁰ × 7⁰ × 3⁰ = ?

5⁰ = 1

7⁰ = 1

3⁰ = 1

1 × 1 × 1 = 1

\(Find \;lcd \;of\; two \;fractions\;5/7\; and\;2/3\\\\lcd(7, 3)\; =\; 21\\21/7\; =3\; -\;additional\;multiplier\;of\;the\;first\;fraction\\\\21/3\;=\;7-\;additional\;multiplier\; of\; the \;second \;fraction\\\\5/7=5*3/7*3=15/21\\\\2/3 =2*7/3*7=14/21\\\\2.)\;Sum\;the\;fractions\;with\;the\;same\;denominator:\\\\15/21+14/21=15+14/21=29/21\)

\(3.)\;Since\;the\;numerator\;is\;greater\;than\;the\;denominator,\;\\convert\;the\;improper\;fraction\;to\;a\;mixed\;number.\)

\(29/21=1*21+8/21=1*21/21 +8/21=1+8/21\)

\(So,\;your\;final\;answer\;is\;1\frac{8}{21}\).

Answer: 3.

Step-by-step explanation:

\(\left \{ {{4x-3y=3 \ |[*(-4)]} \atop {5x-4y=3} \ |[*3]} \right. \ = > \ \left \{ {{-16x+12y=-12} \atop {15x-12y=9}} \right. \ = > \ \left \{ {{y=3} \atop {x=3}} \right.\)

The transformed vertices are;

A''  => (-7, -1)

B''  => (-7, 4)

C''=> (-9,4)

D''=> (-9, -1)

Here, we have,

given that,

we have to translate by (x,y) => (x-5, y+4)

so, the rectangle will be transformed as;

the transformed vertices are;

A' => ( 4-5, 3+4) => (-1, 7)

B' => (9-5, 3+4) => (4,7)

C'=>(9 -5, 5 +4 ) => (4, 9)

D'=> (4 -5, 5+4 ) => (-1, 9)

now, For clockwise rotation of a triangle by 90 degree, then x coordinate is similar to the y coordinate of original point and y coordinate is negative times of x coordinate of original point.

So (x,y) changes to (-y,x).

the transformed vertices are;

A'' => ( 4-5, 3+4) => (-1, 7) => (-7, -1)

B'' => (9-5, 3+4) => (4,7) => (-7, 4)

C''=>(9 -5, 5 +4 ) => (4, 9) => (-9,4)

D''=> (4 -5, 5+4 ) => (-1, 9)=> (-9, -1)

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Answer:

  96°

Step-by-step explanation:

You want the value of angle C in the diagram with two parallel lines and a triangle between them.

The sum of angles in a triangle is 180°, so the missing angle in the triangle is ...

  180° -60° -24° = 96°

Angle C and the one we just found are alternate interior angles with respect to the parallel lines and the transversal that forms those angles. As such, they are congruent:

  C = 96°

<95141404393>

The total value of the furniture sold by Joyce last week is A = $ 6,733.33

Given data ,

Let's call the value of the furniture Joyce sold "x".

She receives 10% commission on the first $1,200 of sales, which is $120:

Commission on first $1,200 = 0.1 * 1200 = 120

The remaining sales that she earns commission on is the difference between the total value of the furniture sold and the first $1,200:

Remaining sales = x - 1200

She earns 15% commission on the remaining sales, which is 0.15 times the remaining sales:

Commission on remaining sales = 0.15 * (x - 1200)

Her total earnings from commission last week was $950. So we can write an equation:

Commission on first $1,200 + Commission on remaining sales = $950

Substituting the expressions we found for the two commissions, we get:

120 + 0.15(x - 1200) = 950

120 + 0.15x - 180 = 950

On simplifying the equation , we get

0.15x - 60 = 950

0.15x = 1010

x = 6,733.33

Hence , the value of the furniture Joyce sold last week was $6,733.33

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Answer:

c..........................

Answer:

B (1, -3)

Step-by-step explanation:

Step 1:  Use the midpoint formula to find the coordinates of the endpoint:

Normally, we find the midpoint of a segment using the midpoint formula, which is given by:

M = (x1 + x2) / 2, (y1 + y2) / 2, where

Since we're solving for the coordinates of an endpoint, we can allow (-5, 7) to be our (x1, y1) point and plug in (-2, 2) for M to find (x2, y2), the coordinates of the endpoint B:

x-coordinate of B:

x-coordinate of midpoint = (x1 + x2) / 2

(-2 = (-5 + x2) / 2) * 2

(-4 = -5 + x2) + 5

1 = x2

Thus, the x-coordinate of the endpoint B is 1.

y-coordinate of B:

y-coordinate of midpoint = (y1 + y2) / 2

(2 = (7 + x2) / 2) * 2

(4 = 7 + x2) -7

-3 = y2

Thus, the y-coordinate of the endpoint B is -3.

Thus, the coordinates of the endpoint B are (1, -3).

Answer:

D

Step-by-step explanation:

sin  (  3pi / 4 )

 =  sin   (   pi   -   pi  /  4   )

 =  sin   (   pi / 4  )

 = 1/root(2)

  =  root(2) / 2

The par value per share before the stock dividend is $5; after the stock dividend, the par value remains $5.

The par value is the entity's stated nominal value.

It does not affect the market value of the shares.

With a stock dividend, the value of the common stock increases by the par value and the number of shares outstanding, making the par value remain the same.

The par value changes when there is a stock split.

Common stock, $5 par value $382,000

Common stock shares = 76,400 shares ($382,000/$5)

Paid-in capital in excess of par-common stock 20,000

Retained earnings 166,000

Total stockholders' equity $568,000

October 1: Retained Earnings $109,960 (7,640 x $14) Stock Dividend Distributable $38,200 Paid-in Capital Capital $71,760

Stock Dividend Distributable $38,200 Common Stock $38,200

Common stock shares = 84,040 shares (76,400 + 7,640)

Common stock value = $420,200

Par value after the stock dividend = $5 ($420,200/84,040)

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Answer:

A

Step-by-step explanation:

(3x - 14)^2

9x^2 - 42x -42x +196

9x^2 - 84 +196 is the correct answer

Answer:

the experimental probability of being struck by lightning in this city is approximately 0.56667%.

Step-by-step explanation:

The experimental probability of an event happening is calculated by dividing the number of times the event occurred by the total number of trials or observations. In this case, the event is being struck by lightning, and the trials are the total number of people in the city.

The number of people struck by lightning is given as 85, and the total population of the city is 1.5 million. To find the experimental probability of being struck by lightning in this city, we can divide the number of people struck by lightning by the total population and multiply by 100% to express the answer as a percentage:

Experimental probability = (number of people struck by lightning / total population) * 100%

Experimental probability = (85 / 1,500,000) * 100%

Experimental probability = 0.0056667 * 100%

Experimental probability = 0.56667%

Therefore, the experimental probability of being struck by lightning in this city is approximately 0.56667%.

Answer:

656

Step-by-step explanation:

Answer:

\(\displaystyle \frac{\pi(5e^4+8e^2-1)}{2e^4}\approx9.526\)

Step-by-step explanation:

This can be solved with either the washer (easier) or the shell method (harder). For the disk/washer method, the slice is perpendicular to the axis of revolution, whereas, for the shell method, the slice is parallel to the axis of revolution.  I'll show how to do it with both:

Shell Method (Horizontal Axis)

\(\displaystyle V=2\pi\int^d_cr(y)h(y)\,dy\)

Radius: \(r(y)=2-y\) (distance from y=2 to x-axis)

Height: \(h(y)=2-(-\ln y)=2+\ln y\) (\(y=e^{-x}\) is the same as \(x=-\ln y\))

Bounds: \([c,d]=[e^{-2},1]\) (plugging x-bounds in gets you this)

Plugging in our integral, we get:

\(\displaystyle V=2\pi\int^1_{e^{-2}}(2-y)(2+\ln y)\,dy=\frac{\pi(5e^4+8e^2-1)}{2e^4}\approx9.526\)

Washer Method (Parallel to x-axis)

\(\displaystyle V=\pi\int^b_a\biggr(R(x)^2-r(x)^2\biggr)\,dx\)

Outer Radius: \(R(x)=2-e^{-x}\) (distance between \(y=2\) and \(y=e^{-x}\))

Inner Radius: \(r(x)=2-1=1\) (distance between \(y=2\) and \(y=1\))

Bounds: \([a,b]=[0,2]\)

Plugging in our integral, we get:

\(\displaystyle V=\pi\int^2_0\biggr((2-e^{-x})^2-1^2\biggr)\,dx\\\\V=\pi\int^2_0\biggr((4-4e^{-x}+e^{-2x})-1\biggr)\,dx\\\\V=\pi\int^2_0(3-4e^{-x}+e^{-2x})\,dx\\\\V=\pi\biggr(3x+4e^{-x}-\frac{1}{2}e^{-2x}\biggr)\biggr|^2_0\\\\V=\pi\biggr[\biggr(3(2)+4e^{-2}-\frac{1}{2}e^{-2(2)}\biggr)-\biggr(3(0)+4e^{-0}-\frac{1}{2}e^{-2(0)}\biggr)\biggr]\\\\V=\pi\biggr[\biggr(6+4e^{-2}-\frac{1}{2}e^{-4}\biggr)-\biggr(4-\frac{1}{2}\biggr)\biggr]\)

\(\displaystyle V=\pi\biggr[\biggr(6+4e^{-2}-\frac{1}{2}e^{-4}\biggr)-\frac{7}{2}\biggr]\\\\V=\pi\biggr(\frac{5}{2}+4e^{-2}-\frac{1}{2}e^{-4}\biggr)\\\\V=\pi\biggr(\frac{5}{2}+\frac{4}{e^2}-\frac{1}{2e^4}\biggr)\\\\V=\pi\biggr(\frac{5e^4}{2e^4}+\frac{8e^2}{2e^4}-\frac{1}{2e^4}\biggr)\\\\V=\pi\biggr(\frac{5e^4+8e^2-1}{2e^4}\biggr)\\\\V=\frac{\pi(5e^4+8e^2-1)}{2e^4}\approx9.526\)

Use your best judgment when deciding on what method you use when visualizing the solid, but I hope this helped!

Answer:

------------------

Simplify in below steps:

Answer:

z = (3/√2) + (1/√2)î = (1/√2) [3 + i] = (2.1213 + 0.7071i)

OR

z = -(3/√2) + i(1/√2) = (1/√2) [-3 + i] = (-2.1213 + 0.7071i)

p = (3/√2) = 2.1213

q = (1/√2) = 0.7071

OR

p = (-3/√2) = -2.1213

q = (1/√2) = 0.7071

Step-by-step explanation:

z = √(4 + 3i)

Let the complex number z be equal to

z = p + qi

So, we can write

z = p + qi = √(4 + 3i)

p + qi = √(4 + 3i)

Square both sides

(p + qi)² = [√(4 + 3i)]²

p² + pqi + pqi + (qi)² = (4 + 3i)

p² + 2pqi + q²i² = 4 + 3i

note that i² = -1

p² + 2pqi - q² = 4 + 3i

(p² - q²) + 2pqi = 4 + 3i

Comparing both sides, and them equating the real parts on both sides to each other and the complex parts to each other

(p² - q²) = 4 (eqn 1)

2pq = 3 (eqn 2)

From eqn 2

p = (3/2q)

p² = (9/4q²)

Substituting this into eqn 1

(9/4q²) - q² = 4

multiplying through by 4q²

9 - 4q⁴ = 16q²

4q⁴ + 16q² - 9 = 0

let q² = x, q⁴ = x²

4x² + 16x - 9 = 0

Solving the quadratic equation

x = 0.5 or -4.5

q² = 4

q² = 0.5 or q² = -4.5

q = √0.5 or √-4.5

q = (1/√2) = (√2)/2 = 0.7071

Or q = i(3/√2) = i(3√2)/2 = 2.1213I

p = (3/2q)

If q = (1/√2) = (√2)/2 = 0.7071

p = (3/√2) = (3√2)/2 = 2.1213

if q = i(3/√2) = i(3√2)/2 = 2.1213I

p = i(1/√2) = i(√2)/2 = 0.7071i

z = p + qi

If q = (1/√2) = (√2)/2 = 0.7071

p = (3/√2) = (3√2)/2 = 2.1213

z = (3/√2) + (1/√2)î = (1/√2) [3 + i]

= 2.1213 + 0.7071i

if q = i(3/√2) = i(3√2)/2 = 2.1213I

p = i(1/√2) = i(√2)/2 = 0.7071i

z = i(1/√2) + [i(3/√2) × i]

z = i(1/√2) - (3/√2)

z = -(3/√2) + i(1/√2)

z = (1/√2) [-3 + i]

z = -2.1213 + 0.7071i

Hope this Helps!!!

Answer:

Step-by-step explanation:

18+11*65=2

Answer:

\(\textsf{1.} \quad y=-\dfrac{1}{30}x^2+\dfrac{1}{2}x+\dfrac{68}{15}\:\:\textsf{ where}\:\:x \geq \dfrac{15-\sqrt{769}}{2}\\\\\quad \textsf{or} \quad y=2\sqrt{x+5}\)

\(\textsf{2.} \quad y=-|x+1|+5\)

Step-by-step explanation:

Method 1 - modelling as a quadratic with restricted domain

Assuming that the points given on the graph are points that the curve passes through, the curve can be modeled as a quadratic with a limited domain.  Please note that as the x-intercept has not been defined on the graph, I am not including this in this first method.

Standard form of a quadratic equation:

\(y=ax^2+bx+c\)

Given points:

Substitute the given points into the equation to create 3 equations:

Equation 1  (-4, 2)

\(\implies a(-4)^2+b(-4)+c=2\)

\(\implies 16a-4b+c=2\)

Equation 2  (-1, 4)

\(\implies a(-1)^2+b(-1)+c=4\)

\(\implies a-b+c=4\)

Equation 3  (4, 6)

\(\implies a(4)^2+b(4)+c=6\)

\(\implies 16a+4b+c=6\)

Subtract Equation 1 from Equation 3 to eliminate variables a and c:

\(\implies (16a+4b+c)-(16a-4b+c)=6-2\)

\(\implies 8b=4\)

\(\implies b=\dfrac{4}{8}\)

\(\implies b=\dfrac{1}{2}\)

Subtract Equation 2 from Equation 3 to eliminate variable c:

\(\implies (16a+4b+c)-(a-b+c)=6-4\)

\(\implies 15a+5b=2\)

\(\implies 15a=2-5b\)

\(\implies a=\dfrac{2-5b}{15}\)

Substitute found value of b into the expression for a and solve for a:

\(\implies a=\dfrac{2-5(\frac{1}{2})}{15}\)

\(\implies a=-\dfrac{1}{30}\)

Substitute found values of a and b into Equation 2 and solve for c:

\(\implies a-b+c=4\)

\(\implies -\dfrac{1}{30}-\dfrac{1}{2}+c=4\)

\(\implies c=\dfrac{68}{15}\)

Therefore, the equation of the graph is:

\(y=-\dfrac{1}{30}x^2+\dfrac{1}{2}x+\dfrac{68}{15}\)

\(\textsf{with the restricted domain}: \quad x \geq \dfrac{15-\sqrt{769}}{2}\)

Method 2 - modelling as a square root function

Assuming that the points given on the graph are points that the curve passes through, and the x-intercept should be included, we can model this curve as a square root function.

Given points:

The parent function is:

\(y=\sqrt{x}\)

Translated 5 units left so that the x-intercept is (0, -5):

\(\implies y=\sqrt{x+5}\)

The curve is stretched vertically, so:

\(\implies y=a\sqrt{x+5} \quad \textsf{(where a is some constant)}\)

To find a, substitute the coordinates of the given points:

\(\implies a\sqrt{-4+5}=2\)

\(\implies a=2\)

\(\implies a\sqrt{-1+5}=4\)

\(\implies 2a=4\)

\(\implies a=2\)

\(\implies a\sqrt{4+5}=6\)

\(\implies 3a=6\)

\(\implies a=2\)

As the value of a is the same for all points, the equation of the line is:

\(y=2\sqrt{x+5}\)

Vertex form of an absolute value function

\(f(x)=a|x-h|+k\)

where:

From inspection of the given graph:

Substitute the given values into the function and solve for a:

\(\implies a|0-(-1)|+5=4\)

\(\implies a+5=4\)

\(\implies a=-1\)

Substituting the given vertex and the found value of a into the function, the equation of the graph is:

\(y=-|x+1|+5\)

Answer:

If two chords intersect in a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

6(3x) = 4(4x + 1)

18x = 16x + 4

2x = 4

x = 2

The perimeter of the rectangle is 62 square units.

The perimeter of a rectangle can be calculated using the formula:

P = 2(L + W)

Where:

L is the length of the rectangle

W is the width of the rectangle

We have:

L = 20 units

W = 11 units

Substituting into the formula:

P = 2(20 + 11)

P = 2(31)

P = 62 square units

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Question in English

What is the perimeter of a rectangle if it is 20 long and 11 wide?

For the system of equations, there are infinitely many solutions exist.

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

The system of equations are,

⇒ - 2x + 2y = 10  .. (i)

⇒ y = x + 5  .. (ii)

Here, From (i) equation,

⇒ - 2x + 2y = 10

Take 2 as common,

⇒ 2 (- x + y) = 10

⇒ - x + y = 5

⇒ y = x + 5

Which is same as equation (ii).

Hence, There are infinitely many solutions exist.

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Answer:

Solution given:

\(f(x)=\frac{x-16}{x^2+6x-40}\)

\(g(x)=\frac{1}{x+10}\)

now

f(x)+g(x)=\(\frac{x-16}{x^2+6x-40}+\frac{1}{x+10}\)....(1)

now

factoring x²+6x-40

we get

x²+10x-4x-40

x(x+10)-4(x+10)

(x+10)(x-4)

now substituting in equation 1 ,we get

f(x)+g(x)=\(\frac{x-16}{(x+10)(x-4)}+\frac{1}{x+10}\)

taking l.c.m

=\(\frac{(x-16)+(x-4)}{(x-10)(x-4)}\)

=now

opening bracket

\(\frac{x-16+x-4}{x²-10x-4x+40}\)

=\(\frac{2x-20}{x²+6x-40}\)

So

answer is :

B. \(\bold b\frac{2x-20}{x^2+6x-40}\)